The present invention relates to a method and apparatus for simulating fields especially electromagnetic fields, particularly useful in the context of analysis of interconnect structures, is presented.
Many problems in engineering, physics and chemistry require solving systems of partial differential equations of the type:                               ∇          →                ⁢                  ·                                    J              →                                      (              k              )                                          +                        ∂                      ρ                          (              k              )                                                ∂          t                      =          S              (        k        )              ;
k being a positive whole number
In this equation (equation 1), J represents a flux of a substance under consideration whose density is given by xcfx81 and S represents some external source/sink of the substance. To mention a few examples:
Electrical engineering:
xcfx81 is the charge density,
J is the current density,
S is the external charge source (recombination, generation, . . . )
Structural engineering
Computational Fluid Dynamics:
xcfx81(i)=u(i) the components of the fluid velocity field,             J              (        i        )              =          P      -                        μ          3                ⁢                  ∇                      ·                          u                              (                i                )                                                    ⁢                  xe2x80x83                ⁢        the        ⁢                  xe2x80x83                ⁢        pressure        ⁢                  xe2x80x83                ⁢        tensor              ,      
    ⁢            S              (        i        )              =                            μ          ρ                ⁢                              ∇            2                    ⁢                      u                          (              i              )                                          +                                    F                          (              i              )                                m                ⁢                  xe2x80x83                ⁢        the        ⁢                  xe2x80x83                ⁢        external        ⁢                  xe2x80x83                ⁢        force              ,      
    ⁢                    ∂        ρ                    ∂        t              ->                            ∂          ρ                          ∂          t                    +                                    ∇            ->                    ⁢                      ·                          u              ->                                      ⁢                  xe2x80x83                ⁢        the        ⁢                  xe2x80x83                ⁢        convective        ⁢                  xe2x80x83                ⁢                  derivative          .                    
The list is not exhaustive and there exist many examples where problems are reformulated in such a form that their appearance is as in equation (1). An important example is the Laplace equation and Poisson equation, in which
{right arrow over (J)}={right arrow over (∇)}"psgr"
is the derivative of a scalar field.
There have been presented a number of methods for solving a set of partial differential equations as given above. All numerical methods start from representing the continuous problem by a discrete problem on a finite set of representative nodes in the domain where one is interested in the solution. In other words a mesh is generated in a predetermined domain. The domain can be almost anything ranging from at least a part of or a cross-section of a car to at least a part of or a cross-section of a semiconductor device. For clarification purposes, the discussion is limited here to two-dimensional domains and two-dimensional meshes. This mesh comprises nodes and lines connecting these nodes. As a result, the domain is divided into two-dimensional elements. The shape of the elements depends amongst others on the coordinate system that is chosen. If for example a Cartesian coordinate system is chosen, the two-dimensional elements are e.g. rectangles or triangles. Using such a mesh, the domain can be introduced in a computer aided design environment for optimization purposes. Concerning the mesh, one of the issues is to perform the optimization using the appropriate amount of nodes at the appropriate location. There is a minimum amount of nodes required in order to ensure that the optimization process leads to the right solution at least within predetermined error margins. On the other hand, if the total amount of nodes increases, the complexity increases and the optimization process slows down or even can fail. Because at the start of the optimization process, the (initial) mesh usually thus not comprise the appropriate amount of nodes, additional nodes have to be created or nodes have to be removed. Adding nodes is called mesh refinement whereas removing nodes is called mesh coarsening. Four methods are discussed. As stated above, for clarification and simplification purposes the xe2x80x98languagexe2x80x99 of two dimensions is used, but all statements have a translation to three or more dimensions.
The finite-difference method is the most straightforward method for putting a set of partial differential equations on a mesh. One divides the coordinate axes into a set of intervals and a mesh is constructed by all coordinate points and replaces the partial derivatives by finite differences. The method has the advantage that it is easy to program, due to the regularity of the mesh. The disadvantage is that during mesh refinement many spurious additional nodes are generated in regions where no mesh refinement is needed.
The finite-box method, as e.g. in A. F. Franz, G. A. Franz, S. Selberherr, C. Ringhofer and P. Markowich xe2x80x9cFinite Boxes-A Generalization of the Finite-Difference Method Suitable for Semiconductor Device Simulationxe2x80x9d IEEE Trans. on Elec. Dev. ED-30, 1070 (1983), is an improvement of the finite-difference method, in the sense that not all mesh lines need to terminate at the domain boundary. The mesh lines may end at a side of a mesh line such that the mesh consists of a collection of boxes, i.e. the elements. However, numerical stability requires that at most one mesh line may terminate at the side of a box. Therefore mesh refinement still generates a number of spurious points. The issue of the numerical stability can be traced to the five-point finite difference rule that is furthermore exploited during the refinement.
The finite-element method is a very popular method because of its high flexibility to cover domains of arbitrary shapes with triangles. The choice in favor of triangles is motivated by the fact that each triangle has three nodes and with three points one can parameterize an arbitrary linear function of two variables, i.e. over the element the solution is written as
"psgr"(x,y)=a+b.x+c.y 
In three dimensions one needs four points, i.e. the triangle becomes a tetrahedron. The assembling strategy is also element by element. Sometimes for CPU time saving reasons, one performs a geometrical preprocessing such that the assembling is done link-wise, but this does not effect the element-by-element discretization and assembling. The disadvantage is that programming requires a lot of work in order to allow for submission of arbitrary complicated domains. Furthermore, adaptive meshing is possible but obtuse triangles are easily generated and one must include algorithms to repair these deficiencies, since numerical stability and numerical correctness suffers from obtuse triangles. As a consequence, mesh refinement and in particular adaptive meshing, generates in general spurious nodes.
The finite-element method is not restricted to triangles in a plane. Rectangles (and cubes in three dimensions) have become popular. However, the trial functions are always selected in such a way that a unique value is obtained on the interface. This restriction makes sense for representing scalar functions "psgr"(x,y) on a plane.
In the box-integration method, each node is associated with an area (volume) being determined by the nodes located at the closest distance from this node or in other words, the closest neighbouring node in each direction. Next, the flux divergence equation is converted into an integral equation and using Gauss theorem, the flux integral of the surface of each volume is set equal to the volume integral at the right hand side of the equation, i.e. equation 1 becomes             ∫              ∂                  xe2x80x83                ⁢                  Ω          n                      ⁢                            J          ->                          (          i          )                    ·                        ⅆ          ->                ⁢        s              ⁢      xe2x80x83    =            ∫              Ω        n              ⁢                  (                              S                          (              i              )                                -                                    ∂                              ρ                                  (                  i                  )                                                                    ∂              t                                      )            ⁢                        ⅆ          n                ⁢        x            
The assembling is done node-wise, i.e. for each node the surface integral is decomposed into contributions to neighboring nodes and the volume integral at the right-hand side is approximated by the volume times the nodal value. The spatial discretization of the equation then becomes             ∑      k        ⁢                  J                  1          ⁢          k                    ⁢                        ∂                      xe2x80x83                    ⁢                      Ω                          1              ⁢              k                                                h                      1            ⁢            k                                =            (                        S          1                      (            i            )                          -                              ∂                          ρ              l                              (                i                )                                                          ∂            t                              )        ⁢    Δ    ⁢          xe2x80x83        ⁢          Ω      1      
The advantages/disadvantages of the method are similar as for the Finite element method because the control volumes and the finite elements are conjugate or dual meshes. Voronoi tessellation with the Delaunay algorithm is often exploited to generate the control volumes.
However, forming and refining the mesh is not the only problem facing the skilled person in the solution of field theory problems. For instance, the on-chip interconnect structure in modern ULSI integrated circuits is a highly complex electromagnetic system. The full structure may connect more than one million transistors that are hosted on a silicon substrate, and containing up to seven metallization layers, and including interconnect splittings, curves, widenings, etc. A structure results with a pronounced three-dimensional character. As a consequence, analytic solution methods have only limited applicability and numerical or computer-aided design methods need to be used. The continuous down-scaling of the pitch implies that parasitic effects become a major design concern. Furthermore, interconnect delay will soon become the main bottleneck for increasing the operation frequencies of the fully integrated circuit. These observations justify an in-depth analysis of the interconnect problem based on the basic physical laws underlying the description of these systems. Whereas in the past it sufficed to extract the parasitic behavior from the low-frequency values of the characteristic parameters, such as the resistance (R), capacitance (C) and inductance (L), knowledge about the modifications of these parameters due to fast variations in time of the fields, i.e. at high frequency, becomes mandatory. A generic method that allows one to obtain the frequency dependence of the characteristic parameters for an interconnect (sub-) system has been a requirement for some time.
The highest frequency in which is currently of interest is 50 GHz, which corresponds to a shortest wavelength of the order of one centimeter. However, this is only a current limit. For most of the interconnects with sub-micron widths the characteristic width (length) of the structure is therefore much smaller than the wavelengths under consideration. In this regime one normally neglects the full displacement current, but this view must be refined depending upon the materials used [H. K. Dirks, Quasi-Stationary Fields for Microelectronic Applications, Electrical Engineering, 79, 145-155, 1996]. Interconnect lines are typically parallel to the axes of a Cartesian grid Manhattan like geometry. Although this is no longer true for widenings and splittings in the lines and the vertical connections, i.e. cylindrical vias, most of the structure can be regarded in a first order approximation as consisting of straight orthogonal lines or bricks. The skin effect becomes important for the upper metallization levels where the width of the structures is larger than the skin depth for aluminum or copper, especially at the high frequency part of the spectrum. Eddy currents play an important role in the lossy semiconductor silicon substrate. It is desirable to formulate the equations for the interconnect system in a language that is familiar to the interconnect-designer community. In particular, variables such as the Poisson field should have their usual meaning. For time-dependent fields it can be achieved by selecting a specific gauge fixing. In particular, in the Coulomb gauge, the Poisson equation remains unaltered. The natural choice for the description of interconnect systems is the one that uses the electric scalar potential and the magnetic vector potential. Small signal analysis (AC analysis) has been a successful tool for extracting compact model parameters for devices [S. E. Laux, Techniques for Small-Signal Analysis of Semiconductor devices, IEEE trans. on computer-aided design, 4, 472-481, 1985]. Recently good results were obtained in using small-signal analysis [S. Jenei, private communication, 2000] for the extraction of compact model parameters for the Hasagawa system [H. Hasegawa et al. IEEE Trans. on Microwave Theory and Techniques vol. MTT-19, 869, 1971] and similar methods are currently exploited for the design of spiral inductors.
Numerical analysis is well known to the skilled person, e.g. xe2x80x9cThe finite element methodxe2x80x9d, Zienkiewicz and Taylor, Butterworth-Heinemann, 2000 or xe2x80x9cNumerical Analysisxe2x80x9d, Burden and Faires, Brooks/Cole, 2001. Conventional finite difference numerical analysis solves three-dimensional field theory problems that contain the magnetic vector potential by superimposing three scalar fields, representing this vector potential, whereby each scalar value is located at a node of a mesh. Finite difference methods convert partial differential equations into algebraic equations for each node based on finite differences between a node of interest and a number of neighbours. These methods introduce three types of errors. Firstly, there is the error caused by solving for a discrete mesh, which is only an approximation to a continuum. The smaller the mesh the higher the accuracy. Secondly, the finite difference methods require an iterative solution, which is terminated after a certain timexe2x80x94this implies a residual error. Thirdly, the superposition of three scalar fields is only an accurate representation of vector fields when the mesh size is so small that moving from one node to the next in one direction is associated with a negligible change in the field values in the other two dimensions. In such a case small changes of dimension in one direction may be considered as if the values of the field in the other two are constant. Where there are strongly varying fields this criterion can only be met where the mesh spacing is very small, i.e. there are a large number of nodes. Computational intensity increases rapidly with the number of nodes. To a certain extent the computational intensity can be reduced by modifying the size of mesh so that a tight mesh is only used where the divergence of the field requires this. However, varying mesh sizes places limitations on the continuity of the solution resulting in unnecessary nodes being created to provide sufficiently gradual changes. Hence, conventionally a large amount of storage space and high-powered computers are required to achieve an accurate result in a reasonable amount of time.
It is an aim of the present invention to provide numerically stable methods and apparatus implementing these methods for simulating (i.e. calculating) field problems, e.g. electromagnetic fields.
It is a further aim of the present invention to provide numerically stable methods and apparatus implementing these methods for simulating (i.e. calculating) field problems, e.g. electromagnetic fields which requires less storage space and preferably less computational intensity.
The present invention provides a consistent solution scheme for solving field problems especially electromagnetic modeling that is based upon existing semiconductor techniques. A key ingredient in the latter ones is the numerical solutions method based on a suitable finite difference method such as the Newton-Raphson technique for solving non-linear systems. This technique requires the inversion of large sparse matrices, and of course numerical stability demands that the inverse matrices exist. In particular, the finite difference matrix, e.g. a Newton-Raphson matrix should be square and non-singular. The present invention provides a generic method for solving field problems, e.g. simulating electromagnetic fields, and is designed for numerical stability, in particular the solution of partial differential equations by numerical methods.
It is an aspect of the invention that it is recognized that in order to obtain a consistent discretization scheme, meaning leading to numerical fit calculations, a dummy transformation field, also denoted gauge transformation field or auxiliary gauge field can be introduced as a dummy field and can ease computation. The dummy field can be introduced due to the non-uniqueness of the electric and magnetic potentials describing the underlying physical phenomena.
It is an aspect of the invention that it is recognized that in order to obtain a consistent discretization scheme special caution is taken in the translation of the continuous field equations onto the discrete lattice, comprising of nodes and links.
With the generic method high-frequency parasitic effects and the frequency dependence of the characteristic parameters for an interconnect (sub-) system can be studied but the method is not limited thereto.
The present invention provides a method for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the method comprising:
directly solving the field equations modified by addition of a dummy field by numerical analysis, and
outputting at least one parameter relating to a physical property of the system.
The method can be formalized as follows: a method for simulating fields in or about a device, said method comprising the steps of:
modifying the set of field equations expressed in terms of the vector potential of said fields to a set of modified field equations expressed in terms of the vector potential of said inductive fields and a dummy field; and
directly solving the set of modified field equations in order to obtain the vector potential and said dummy field. The output of the method is a field related parameter of the device, e.g. an electromagnetic parameter of the device such as a field strength, a resistivity, an inductance, a magnetic field strength, an electric field strength, an energy value. The field equations of the above method may be the Maxwell equations. The dummy field is preferably a scalar field.
The present invention also provides a method for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell""s field equations of which the following is a representation:                               ∇                      xc3x97                          (                                                1                  μ                                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xc3x97                    A                                                              )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  J          ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      ε            ⁢                          xe2x80x83                        ⁢                          ∂                              ∂                t                                      ⁢                          xe2x80x83                        ⁢                          (                                                ∇                  V                                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                    A                                                        ∂                    t                                                              )                                                          (        1        )                                          ∇                      ·            A                          ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        0                            (        2        )                                          -                      ∇                          (                              ε                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  V                                            )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        ρ                            (        3        )                                E        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                              -                          ∇                              xe2x80x83                            ⁢              V                                ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                                    ∂              A                                      ∂              t                                                          (        4        )                                B        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  ∇                      xc3x97            A                                              (        5        )            
Where J and xcfx81 are generic functions of the fields, i.e.
J=J(E,B,t)xe2x80x83xe2x80x83(6) 
xcfx81=xcfx81(E,B,t)xe2x80x83xe2x80x83(7) 
the method comprising:
directly solving the field equations modified by addition of a dummy field by numerical analysis, the dummy field removing a singularity in the numerical analysis, and
outputting at least one parameter relating to a physical property of the system.
The physical property may be any of the following non-limiting list: an electric current, a current density, a voltage difference, an electric field value, a plot of an electric field, magnetic field value, a plot of a magnetic field, a resistance or a resistivity conductance or a conductivity, a susceptance or a suceptibility, an inductance, an admittance, a capacitance, a charge, a charge density, an energy of an electric or magnetic field, a permittivity, a heat energy, a noise level induced in any part of a device caused by electromagnetic fields, a frequency.
The above methods also include a step refining a mesh used in the numerical analysis in accordance with an embodiment of the present invention.
The present invention may provide an apparatus for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the apparatus comprising: means for solving by numerical analysis a modification of the field equations, the modification being an addition of a dummy field, and means for outputting at least one parameter relating to a physical property of the system.
The present invention may also provide an apparatus for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell""s field equations of which the following is a representation:                               ∇                      xc3x97                          (                                                1                  μ                                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xc3x97                    A                                                              )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  J          ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      ε            ⁢                          xe2x80x83                        ⁢                          ∂                              ∂                t                                      ⁢                          xe2x80x83                        ⁢                          (                                                ∇                  V                                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                    A                                                        ∂                    t                                                              )                                                                        ∇                      ·            A                          ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        0                                          -                      ∇                          (                              ε                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  V                                            )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        ρ                                E        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                              -                          ∇                              xe2x80x83                            ⁢              V                                ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                                    ∂              A                                      ∂              t                                                              B        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  ∇                      xc3x97            A                              
where
J=J(E,B,t) 
xcfx81=xcfx81(E,B,t) 
the apparatus comprising:
means for directly solving the field equations modified by addition of a dummy field by numerical analysis, the dummy fieled being added to remove a singularity during the numerical analysis, and
means for outputting at least one parameter relating to a physical property of the system.
The present invention may include a data structure for use in numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the field equations being modified by addition of a dummy field, wherein the data structure comprises the simulation of the physical system as a representation of an n-dimensional mesh in a predetermined domain of the physical system, the mesh comprising nodes and links connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2n nodes, the data structure being stored in a memory and comprising representations of the nodes and the links between nodes, the data structure also including definitions of a parameter of the dummy field associated with the nodes of the mesh.
A data structure for use in numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell""s field equations of which the following is a representation:                               ∇                      xc3x97                          (                                                1                  μ                                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xc3x97                    A                                                              )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  J          ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      ε            ⁢                          xe2x80x83                        ⁢                          ∂                              ∂                t                                      ⁢                          xe2x80x83                        ⁢                          (                                                ∇                  V                                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                    A                                                        ∂                    t                                                              )                                                                        ∇                      ·            A                          ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        0                                          -                      ∇                          (                              ε                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  V                                            )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        ρ                                E        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                              -                          ∇                              xe2x80x83                            ⁢              V                                ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                                    ∂              A                                      ∂              t                                                              B        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  ∇                      xc3x97            A                              
where
J=J(E,B,t) 
xcfx81=xcfx81(E,B,t) 
the field equations being modified by addition of a dummy field, wherein the data structure comprises the simulation of the physical system as a representation of an n-dimensional mesh in a predetermined domain of the physical system, the mesh comprising nodes and links connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2n nodes, the data structure being stored in a memory and comprising representations of the nodes and the links between the nodes, the data structure also including definitions of the vector potential A associated with the links of the mesh.
The present invention also includes a program storage device readable by a machine and encoding a program of instructions for executing any of the methods of the present invention.
The present invention also includes a computer program product for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the computer program product comprising:
code for solving the field equations modified by addition of a dummy field by numerical analysis, and
code for outputting at least one parameter relating to a physical property of the system.
The present invention also includes a computer program product for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell""s field equations of which the following is a representation:                               ∇                      xc3x97                          (                                                1                  μ                                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xc3x97                    A                                                              )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  J          ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      ε            ⁢                          xe2x80x83                        ⁢                          ∂                              ∂                t                                      ⁢                          xe2x80x83                        ⁢                          (                                                ∇                  V                                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                    A                                                        ∂                    t                                                              )                                                                        ∇                      ·            A                          ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        0                                          -                      ∇                          (                              ε                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  V                                            )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        ρ                                E        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                              -                          ∇                              xe2x80x83                            ⁢              V                                ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                                    ∂              A                                      ∂              t                                                              B        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  ∇                      xc3x97            A                              
where
J=J(E,B,t) 
xcfx81=xcfx81(E,B,t) 
the computer program product comprising:
code for solving the field equations modified by addition of a dummy field by numerical analysis, and
code for outputting at least one parameter relating to a physical property of the system.
The present invention also includes a method for numerical analysis of a simulation of a physical system, comprising: transmitting from a near location a description of the physical system to a remote location where a processing engine carries out any of the method in accordance with the present invention, and receiving at a near location at least one physical parameter related to the physical system.
The modified field equations are:                                           ∇                          xc3x97                              (                                                      1                    μ                                    ⁢                                      xe2x80x83                                    ⁢                                      ∇                                          xc3x97                      A                                                                      )                                              ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      γ            ⁢                          xe2x80x83                        ⁢                          ∇                              xe2x80x83                            ⁢              χ                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  J          ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                      ε            ⁢                          xe2x80x83                        ⁢                          ∂                              ∂                t                                      ⁢                          xe2x80x83                        ⁢                          (                                                ∇                                      xe2x80x83                                    ⁢                  V                                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                    A                                                        ∂                    t                                                  ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                                      ∂                                          ∇                                              xe2x80x83                                            ⁢                      χ                                                                            ∂                    t                                                              )                                                          (        8        )                                                      ∇                          ·              A                                ⁢                      xe2x80x83                    +                      xe2x80x83                    ⁢                                    ∇              2                        ⁢                          xe2x80x83                        ⁢            χ                          ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        0                            (        9        )                                          -                      ∇                          (                              ε                ⁢                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  V                                            )                                      ⁢                  xe2x80x83                =                  xe2x80x83                ⁢        ρ                            (        10        )                                E        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                              -                          ∇                              (                                  V                  ⁢                                      xe2x80x83                                    +                                      xe2x80x83                                    ⁢                                                            ∂                      χ                                                              ∂                      t                                                                      )                                              ⁢                      xe2x80x83                    -                      xe2x80x83                    ⁢                                    ∂              A                                      ∂              t                                                          (        11        )                                B        ⁢                  xe2x80x83                =                  xe2x80x83                ⁢                  ∇                      xc3x97                          (                              A                ⁢                                  xe2x80x83                                +                                  xe2x80x83                                ⁢                                  ∇                                      xe2x80x83                                    ⁢                  χ                                            )                                                          (        12        )            
where xcex3 is non-zero a scaling factor, which guarantees matching of dimensions.
In the present invention the introduction of the dummy field represented by "khgr" preferably does not modify the vector potential A found from the solution of the modified field equations when compared with the vector potential found from solution of the unmodified field equations. The accuracy of the method may be checked by comparing known algebraic solutions of simple fields with the solution of the method according to the present invention.
In the method the step of directly solving the set of modified field equations is performed by discretizing the set of modified field equations onto a mesh with nodes and links between said nodes. For example, the mesh can be a Cartesian mesh. In particular, in the method the vector potential is defined on the links of the mesh. The advantage of associating a field vector with the links and not with the nodes results from the fact that links define a direction given inherently by the form of the mesh. Hence, a field vector field is associated with an atomic vector element of the mesh. This is a more accurate simulation than using the superposition of scalar fields to simulate a field vector, the present invention makes advantageous use of vector elements in the mesh to solve the field equations more accurately. This reduces the number of nodes required to achieve a certain accuracy. This also means that the amount of memory required is reduced as well as speeding up the calculation time.
In the method the dummy field is also defined on the nodes of the mesh as it is a scalar field. That is in the finite difference method the nodes are used as the reference points for values of the dummy field. In the method other terms in the modified field equations are expressed in terms of the vector potential and the dummy field. In the method the curlxe2x80x94curl operation on a vector potential on a link is expressed in function of the vector potential on the link and the vector potentials on neighboring links of this link. The curlxe2x80x94curl operation on a vector potential on a link is expressed in function of the vector potential on the link and the vector potentials on links, defined by wings with said link.
In the method the step of directly solving can exploit a Newton-Raphson procedure for solving nonlinear equations. In this case it is preferred to select the dummy field in order to have square non-singular matrices in the Newton-Raphson procedure.
In the method the boundary conditions may be determined by solving a Maxwell equation in a space with 1 dimension less than the space in which the original field equations are solved.
In a further aspect of the invention a method, i.e. the so-called Cube-Assembling Method (CAM), is disclosed for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n-1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements. This method may be advantageously combined with other embodiments of the invention for the solution of field theory equations. The domain can be almost anything ranging from at least a part of a car to at least a part of a semiconductor device. For clarification purposes, the present invention will be described with reference to two-dimensional domains and two-dimensional meshes but the present invention is not limited thereto. The shape of the elements depends amongst others on the coordinate system, which is chosen. By applying a mesh on a domain, the domain can be introduced in a computer aided design environment for optimization purposes. Concerning the mesh, one of the issues is to perform the optimization using the appropriate amount of nodes at the appropriate location. There is a minimum amount of nodes required in order to ensure that the optimization process leads to the right solution at least within predetermined error margins. On the other hand, if the total amount of nodes increases, the complexity increases and the optimization process slows down or even can fail. Because at the start of the optimization process, the (initial) mesh usually thus not comprise the appropriate amount of nodes, additional nodes have to be created or nodes have to be removed during the optimization process. Adding nodes is called mesh refinement whereas removing nodes is called mesh coarsening. The method of the present invention succeeds in adding or removing nodes locally. The assembling is done over the elements, being e.g. squares or cubes or hypercubes dependent of the dimension of the mesh. Like the finite-box method, the CAM method is easy to program, even in higher dimensions. However, the CAM method does not suffer from the restriction that only one line may terminate at the side of a box.
According to this aspect of the invention, a method is disclosed for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n-1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2n nodes, said method comprising at least the steps of:
creating a first additional node inside at least one of said first elements by completely splitting said first element in exactly 2n n-dimensional second elements in such a manner that said first additional node forms a corner node of each of said second elements which results in the replacement of said first element by said 2n n-dimensional second elements; and
creating a second additional node inside at least one of said second elements by completely splitting said second element in exactly 2n n-dimensional third elements in such a manner that said second additional node forms a corner node of each of said third elements which results in the replacement of said second element by said 2n n-dimensional third elements.
In an embodiment of the invention after the mesh is locally refined, this mesh is locally coarsened.
In another embodiment of the invention, the refinement is based on an adaptive meshing strategy.
In another aspect of the invention, a program storage device is disclosed storing instructions that when executed by a computer perform the method for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n-1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2n nodes, said method comprising at least the steps of:
creating a first additional node inside at least one of said first elements by completely splitting said first element in exactly 2n n-dimensional second elements in such a manner that said first additional node forms a corner node of each of said second elements which results in the replacement of said first element by said 2n n-dimensional second elements; and
creating a second additional node inside at least one of said second elements by completely splitting said second element in exactly 2n n-dimensional third elements in such a manner that said second additional node forms a corner node of each of said third elements which results in the replacement of said second element by said 2n n-dimensional third elements.
In an aspect of the invention a method is disclosed for optimizing of a predetermined property of a n-dimensional structure, said method comprising the steps of:
creating a n-dimensional mesh on at least a part of said structure; said mesh containing nodes and n-1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2n first element;
refining said n-dimensional mesh by creating a first additional node inside at least one of said first elements by completely splitting said first element in exactly 2n n-dimensional second elements in such a manner that said first additional node forms a corner node of each of said second elements which results in the replacement of said first element by said 2n n-dimensional second elements;
further refining said n-dimensional mesh by creating a second additional node inside at least one of said second elements by completely splitting said second element in exactly 2n n-dimensional third elements in such a manner that said second additional node forms a corner node of each of said third elements which results in the replacement of said second element by said 2n n-dimensional third elements; and
where said n-dimensional mesh is used to create an improved structure.
In an embodiment of the invention said structure improvements are based on extracting said property from structure characteristics, determined at a subset of said nodes of said mesh.
In a further embodiment of the invention said structure characteristics are determined by solving the partial differential equations, describing the physical behavior of said structure, on said mesh.
The present invention will now be described with reference to the following drawings.